We investigate the challenge of designing robust external excitations to control and synchronize a population of non-interacting homotypic harmonic oscillators, specifically, theta neurons. The Theta model emulates the bursting behavior observed in spiking cells, characterized by periodic oscillations in their membrane electric potential.Our approach involves formulating this optimization task as an optimal mean-field control problem for the linear continuity/Fokker-Planck equation in the space of probability measures. To address this problem numerically, we employ an indirect deterministic descent method, leveraging an exact representation of the increment of the objective functional.As a main contribution, we delve into practical aspects in the implementation of the proposed method and expose several results of numerical experiments.

We study the problem of designing “robust” external excitations for control and synchronization of an assembly of homotypic harmonic oscillators representing so-called theta neurons. The model of theta neuron (Theta model) captures, in main, the bursting behavior of spiking cells in the brain of biological beings, enduring periodic oscillations of the electric potential in their membrane. Our task is to find an external stimulus (control), which steers all neurons of a given population to their desired phases (i.e., excites/slows down its spiking activity) with the highest probability. Our methodology is the following: The optimization problem at hand is formulated as an optimal mean-field control problem for the local continuity equation in the space of probability measures. To solve this problem numerically, we propose an indirect deterministic descent method based on an exact representation of the increment (infinite-order variation) of the objective functional. We illustrate the modus operandi of the proposed method discuss some aspects of its practical realization and provide some results of numerical experiments.